CoSynFlow: Conformal Symplectic Neural Flows for Cross-System Prediction of Dissipative Hamiltonian Dynamics

2026-08-01

AI Summary

Existing neural operator methods struggle to preserve the conformal symplectic geometric structure inherent in dissipative Hamiltonian systems, leading to inaccurate long-term predictions. This work proposes CoSynFlow, the first framework to explicitly embed conformal symplectic structure into a neural flow architecture. By composing symplectic shear mappings with explicit conformal scaling, and integrating Hamiltonian descriptor conditioning alongside physics-informed training, CoSynFlow constructs a structure-preserving continuous-time dynamical model. The method achieves high-fidelity cross-system predictions without retraining, simultaneously attaining minimal trajectory error and machine-precision-level conservation of geometric structure over long-term evolution.

Abstract

Learning solution operators for differential equations is a central problem in scientific machine learning. However, many neural operator methods optimize prediction accuracy without explicitly enforcing the geometric structure of the dynamics. Structure-preserving models such as SympNets and Symplectic Neural Flows address this issue for conservative Hamiltonian systems by preserving the symplectic form. In dissipative Hamiltonian systems with conformal symplectic structure, however, the symplectic form evolves according to a conformal factor determined by the dissipation. We propose CoSynFlow, a conformal symplectic neural flow for learning continuous-time solution maps of dissipative Hamiltonian dynamics. CoSynFlow composes symplectic shear maps with explicit conformal scaling, preserving the conformal symplectic structure by construction. By conditioning it on a finite-dimensional Hamiltonian descriptor and the dissipation parameter, a single trained model predicts solution maps for unseen systems without retraining. CoSynFlow keeps the structure error at machine precision, attains the lowest long-horizon error, and admits physics-informed training.

Problem

• dissipative Hamiltonian systems
• conformal symplectic structure
• neural operators
• geometric structure preservation
• solution operators

Innovation

• conformal symplectic
• neural flows
• dissipative Hamiltonian systems
• structure-preserving learning
• solution operators